*(English)*Zbl 1047.34030

The authors investigate oscillatory properties of the linear Hamiltonian system

where $A,B,C$ are $n\times n$-matrices with continuous entries for $t\in [{t}_{0},\infty )$, $B,C$ are symmetric and the matrix $B$ is supposed to be positive definite. Using a transformation which preserves the oscillatory nature of transformed systems, system (*) is transformed into the “diagonal off” system ${y}^{\text{'}}=\tilde{B}\left(t\right)z$, ${z}^{\text{'}}=-\tilde{C}\left(t\right)y$ with $\tilde{B}$ positive definite, hence this system is equivalent to the second-order vector-matrix Sturm-Liouville differential equation

The main result of the paper is formulated under the assumption that the matrix $\tilde{B}\left(t\right)-I$ is nonnegative definite ($I$ denotes the identity matrix). Under this assumption, system (**) is a Sturmian majorant of the system ${y}^{\text{'}\text{'}}+\tilde{C}\left(t\right)y=0$. So, oscillation of the last system implies oscillation of (**) and hence, in turn, also of (*). From this point of view, the results of the paper are very close to those presented in the paper of *G. J. Butler, L. H. Erbe* and *A. B. Mingarelli* [Trans. Am. Math. Soc. 303, 263–282 (1987; Zbl 0648.34031)].